NEW ALBANY, Ind. — A white folding table is arranged at an angle just outside the vestibule for the New Albany High School basketball offices and locker rooms, with a single red plastic chair situated behind it. Everyone knows what comes next. That’s why they’ve been waiting for about a half-hour, dozens of them, a queue stretching to the double doors at the opposite end of the hallway: men, women and children with their pens and smartphones and expectant stares and murmurs. Just before 8 p.m., Romeo Langford walks into sight. He drops his backpack, takes a seat in the red chair and, as usual, does everything that’s asked of him.

The most coveted uncommitted high school player in the country has just competed in a game broadcast on ESPNU. He did a pre-tipoff interview. He aggravated a sore left wrist early on and played more or less one-handed for a while. His calf muscle locked up and trainers had to massage out the cramp. He took a shot to his nose, which required...